Chromatic dispersion measurement

ABSTRACT

In one method, two light signals, of the same optical frequency, but having orthogonal states of polarization, are transmitted through an optical device and the mean signal delay of each of the light signals is measured. Calculations, based upon disclosed relationships, provide the polarization-independent delay (τ 0 ) through the optical device based upon the mean signal delays (τ g1  and τ g(−1 )) of each of the light signals. By comparing τ 0  at adjacent wavelengths, the chromatic dispersion of the optical device can be accurately measured even in the presence of PMD. In a second, similar method, four light signals of non-degenerate polarizations states that span Stokes space are utilized. In a modification of the above-described methods based on the measurement of pulse delays, the methods are adapted to the measurement of phase delays of sinusoidally modulated signals.

[0001] This application is a continuation in part of co-pending U.S.patent application Ser. No. 09/520,537, filed Mar. 08, 2000; the subjectmatter of which is incorporated herein by reference.

BACKGROUND OF THE INVENTION

[0002] This invention relates to optical measurements, and particularlyto the measurement of the chromatic dispersion of optical devices,particularly optical fibers used in telecommunication networks.

[0003] Chromatic dispersion is the difference in time delay thatdifferent frequencies (wavelengths) experience when being transmittedthrough an optical component or optical fiber. This phenomenon is causedby the frequency-dependence of the refractive index (n) of the materialand the waveguide dispersion, which is related to the design of theoptical fiber. Polarization mode dispersion (PMD) is thepolarization-dependence of the time delay through an optical componentor an optical fiber. Polarization mode dispersion and chromaticdispersion are related, in that they both reflect time delays through anoptical fiber component. PMD is discussed first.

[0004] PMD is a distortion mechanism (like chromatic dispersion) thatcauses optical devices, such as single-mode fibers, optical switches andoptical isolators, to distort transmitted light signals. The relativeseverity of PMD (which is a function of the wavelength of thetransmitted light) has increased as techniques for dealing withchromatic dispersion have improved, transmission distances haveincreased, and bit rates have increased. Negative effects of PMD appearas random signal fading, increased composite second order distortion andincreased error rates.

[0005] PMD is due to differential group delay caused by geometricalirregularities and other sources of birefringence in the transmissionpath of the optical device. For example, a single-mode fiber (SMF) isideally a homogeneous medium supporting only one mode. In practice, itsupports two propagation modes with orthogonal polarizations. When alightwave source transmits a pulse into a SMF cable, the pulse energy isresolved onto the principal states of polarization of the fiber. The twogroups of pulse energy propagate at different velocities and arrive atdifferent times causing pulse broadening and signal distortion.

[0006] The PMD of a fiber is commonly characterized by two specificorthogonal states of polarization called the principal states ofpolarization (PSPs) and the differential group delay (DGD) between them.This can be described at an optical angular frequency, ω, by the3-component Stokes vector, {right arrow over (Ω)}=Δτ{right arrow over(q)}, where {right arrow over (q)} is a unit Stokes vector pointing inthe direction of the faster PSP, and the magnitude, Δτ , is the DGD.Typical DGD values encountered in transmission systems range between 1(picosecond) ps and 100 ps.

[0007] Known methods for determining PMD vectors include the JonesMatrix Eigenanalysis (JME) technique and the Müller Matrix Method (MMM).Each of these techniques uses a tunable, continuous-wave laser and apolarimeter to measure the output polarization states for two (or three)different input polarization launches at two optical frequencies. ThePMD vector is then calculated for the midpoint frequency. In addition todetermining the output PMD vector, the Müller Matrix Method determinesthe rotation matrix of the fiber at each frequency and thus the inputPMD vector can be calculated.

[0008] Measurements of chromatic dispersion of optical components,spooled fiber, or installed fiber are important for predicting howsevere the pulse distortion (and associated penalties) will be aftertransmission through the optical element(s). Chromatic dispersion isoften measured with the modulation phase-shift method (B. Costa, et al.,Journal of Quantum Electronics, Vol. 18, pp. 1509-1515, 1982). In thismethod, light from a tunable laser is modulated (usually with a sinewave at 1 to 3 MHz frequency) and launched into the optical element. Themean signal delay at the output of the optical element is measured usinga network analyzer by referencing to the input. By measuring the delaysfor two frequencies, the chromatic dispersion at the average of the twofrequencies can be obtained by dividing the change in delay by thechange in frequency. The modulation phase-shift method is conventionallyemployed without control of the polarization of the signals launchedinto the optical element. For an element with no PMD, not controllingthe input polarizations does not change the time delays measured or thecalculated chromatic dispersion. But for birefringent elements orelements with PMD, the mean signal delays measured at the output willdepend on the launched polarization of the signals. Therefore, thechromatic dispersion and the PMD both contribute to the time delay andcannot be easily separated.

[0009] The invention disclosed in this application details a method foraccurately measuring the chromatic dispersion of an optical device inthe presence of PMD. The chromatic dispersion measured is the intrinsicor polarization-averaged chromatic dispersion. Other known methods formeasuring chromatic dispersion tend to give inaccurate results if thedevices have PMD.

SUMMARY OF THE INVENTION

[0010] At the input of an optical device, typically an optical fiberlink within an optical fiber telecommunications network, two differentlight signals, of the same optical frequency, but having differentstates of polarization that are orthogonal in Stokes space, aretransmitted along the fiber and the mean signal delay of each of thelight signals is measured. By repeating the mean signal delaymeasurement at multiple optical frequencies (i.e., at a differentoptical frequency for each set of time delay measurements),determination can be made of the first and higher-order intrinsic(polarization-averaged) chromatic dispersion of the device beingmeasured. A method involving launching four test signals is alsodisclosed, where the launch polarizations are not required to beorthogonal.

DESCRIPTION OF THE DRAWINGS

[0011]FIG. 1 is a schematic illustration of a system for implementingthe present invention;

[0012]FIG. 2 is similar to FIG. 1 but showing a more specific example ofa suitable system for implementing the invention;

[0013]FIG. 3 is a graph showing the wavelength dependence of the signaldelay of each of four fixed input polarization signals transmittedthrough a single mode fiber; and

[0014]FIG. 4 is a graph showing the wavelength dependence of theintrinsic (polarization-averaged) group delay of the fiber obtainedusing the data of FIG. 3.

DESCRIPTION OF PREFERRED EMBODIMENTS

[0015] Preliminary Discussion of PMD Measurements

[0016] As above-described, PMD is usually (and herein) described as athree dimensional vector having a magnitude and direction, i.e., havingthree components. Both an input and an output PMD vector can bedetermined for any optical device. The PMD vectors are a function of thewavelength of the light transmitted through the device. A method formaking measurements of an optical device and determining the threecomponents of the various PMD vectors is first described. Themeasurement and calculations made are based upon the known (Mollenauer,L. F. and Gordon, J.P., Optics Lett., Vol. 19, pp. 375-377, 1994)relationship, $\begin{matrix}{\tau_{g} = {\tau_{0} - {\frac{1}{2}{\overset{\rightarrow}{s} \cdot \overset{\rightarrow}{\Omega}}}}} & (1)\end{matrix}$

[0017] where {right arrow over (s)} and {right arrow over (Ω)} are,respectively, the normalized Stokes vectors of the light and the(unnormalized) PMD vector at the fiber input. Equation (1) describes thepolarization dependence of the mean signal delay, τ_(g), through thefiber as defined by the first moment of the pulse envelope in the timedomain with τ₀ being a polarization averaged or common group delaycomponent. Eq. (1) assumes that τ₀ and {right arrow over (Ω)}(ω) do notvary significantly over the bandwidth of the signal. The(polarization-averaged) chromatic dispersion can be determined by thefrequency dependence of τ₀.

[0018] The definition of mean signal delays involving “moments” of theoutput signals is known and described, for example, in the afore-citedpublication by Mollenauer and Gordon as well as by Elbers, J. P., et al,“Modeling of Polarization Mode Dispersion in Single Mode Fibers,”Electr. Lett., Vol. 33, pp. 1894-1895, October 97; Shieh, W., “PrincipalStates of Polarization for an Optical Pulse,” IEEE Photon. Technol.Lett; Vol. 11, No. 6, p. 677, June 99; and Karlsson, M., “PolarizationMode Dispersion-Induced Pulse Broadening in Optical Fibers,” Opt. Lett.,Vol. 23, pp. 688-690, '98.

[0019] More precisely, mean signal delay, τ_(g), is expressed as thedifference of the normalized first moments at fiber output and input,$\begin{matrix}{{\tau_{g} = \frac{{W_{1}(z)} - {W_{1}(0)}}{W}},} & (2)\end{matrix}$

[0020] where z is the distance of propagation in the fiber. HereW=ƒdt{right arrow over (E)}^(†){right arrow over (E)}=ƒdω{tilde over(E)}^(†){tilde over (E )}is the energy of the signal pulse representedby the complex field vector {right arrow over (E)}(z,t) with Fouriertransform {tilde over (E)}(z,ω), and W₁(z)=ƒdtt{right arrow over(E)}^(†){right arrow over (E)}=jƒdω{tilde over (E)}^(†){tilde over (E)}ωis the first moment.

[0021] Because, in equation (1), there are four unknowns, i.e., τ₀(fiber intrinsic or common group delay) and the three components of thePMD input vector, {right arrow over (Ω)}, four measurements arenecessary to determine {right arrow over (Ω)} and τ₀ at a givenwavelength. Each set of measurements comprises launching a set of fourlight signals of respectively different polarization states into thefiber and measuring the mean signal delay of each of the signals byknown means.

[0022] A variety of different light polarization states can be used, butfor simplicity of the mathematical analysis, a convenient set ofpolarization states are those coinciding with the Poincaré sphere axes,Ŝ₁, Ŝ⁻¹, Ŝ₂, and Ŝ₃, i.e., three different linear polarization states(where Ŝ₁ and Ŝ⁻¹ are orthogonal polarization states) and a circularpolarization state. (A “Poincaré sphere” being a graphicalrepresentation of all possible polarization states on a surface of asphere where each point on the sphere represents a differentpolarization form; see, for example, W. Shurcliff, Polarized Light:Production and Use, p. 16, Harvard University Press, Cambridge, Mass.,1962. A specific example of selected polarization states is providedhereinafter.)

[0023] By identifying the measured mean signal delays for each polarizedstate as τ_(g1), τ_(g(−1)), τ_(g2), and τ_(g3) respectively, thecalculated components from equation (1) are as follows: $\begin{matrix}{\tau_{0} = {\frac{1}{2}\left( {\tau_{g1} + \tau_{g{({- 1})}}} \right)}} & (3)\end{matrix}$

[0024] and the three components of the fiber input PMD Stokes vector Qare:

Ω₁=2(τ₀−τ_(g1))

Ω₂=2(τ₀−τ_(g2))  (4)

Ω₃=2(τ₀−τ_(g3))

[0025] Here, τ₀ is determined directly from the measured mean signaldelays of the two orthogonally polarized test light signals (e.g., Ŝ₁and Ŝ⁻¹). Described hereinafter is how the above mathematicalrelationships, for the “special” Poincaré sphere axial polarizations,can be generalized to any four non-degenerate input polarization states.Also described hereinafter is how, from the measurement of τ₀ atdifferent optical frequencies, chromatic dispersion is determined.

[0026] The basic measuring technique is illustrated in FIG. 1, whichshows a test system including a source 10 of modulated light at a singleoptical frequency and a polarization controller 12 for adjusting thepolarization of the light from the source 10 to successively differentpolarization states, which are successively introduced into andtransmitted through an optical device 14 under test, e.g., a length ofoptical fiber wound on a coil. The light signals exiting the test device14 are successively fed into an apparatus 16 for measuring the meansignal delay for each test signal transmitted through the device. Themeasured data provides, by equation (3) (using orthogonally polarizedtest signals), the polarization-averaged delay, and, by equation 4,using four test signals, the input PMD Stokes vector of the device undertest at a single optical frequency. The measurements are repeated atsuccessively different optical frequencies for determining the frequencydependence of these factors. Also, the data taken at multiplefrequencies allows the determination of the chromatic dispersion of thedevice under test. This is described hereinafter.

[0027] Apparatus for performing each of the functions indicated in FIG.1 are generally known, and different specific arrangements can bedevised by persons of skill. FIG. 2 shows one such arrangement.

[0028] Light of a desired wavelength is provided by a wavelength tunablelaser 20 (Hewlett-Packard 81682A). A directional coupler 22 (EtekSWBC2201PL213) splits the light from the laser into two parts, one ofwhich enters a wavemeter 24 (Hewlett-Packard 86120B) that measures thewavelength of the light, the other of which is coupled to an opticalmodulator (MOD) 26 (Lucent Technologies X2623C). An electrical signalfrom an oscillator 28 (1 GHz) (part of Hewlett-Packard 83420A) is alsoconnected to the modulator. The optical modulator imposes a sinusoidalmodulation on the optical power of the light. This modulated signal isamplified by an optical amplifier (A) 30 (Lucent TechnologiesW1724CDDAD), passed through a tunable optical filter 32 (JDS FitelTB1500B) tuned to the wavelength of the signal and enters a linearpolarizer 34 (Etek FPPD2171LUC02). A polarization controller (PC1) 36(Fiber Control Industries FPC-1) is used to adjust the polarization ofthe signal so that it nearly matches the polarization of the linearpolarizer 34. This polarizer 34, which is not essential to themeasurement, serves to make the results of the measurement insensitiveto polarization changes occurring in the components and in the fiberpreceding the polarizer. After the polarizer 34, the signal passesthough a launch polarizer box 40. This box 40, which functions byswitching polarizers into or out of the signal path, selectivelymodifies the signal polarization from its input polarization to one offour output polarization states.

[0029] Three of those states are provided by putting an appropriatepolarizer in the beam path while the fourth state is provided byswitching all of the polarizers out of the beam path and using the inputsignal polarization unmodified. This input polarization can be adjustedto a desired state using polarization controller (PC2) 42 (Fiber ControlIndustries FPC-1). Alternatively, the fourth polarization could beprovided by a fourth switchable polarizer in launch polarizer box 40.After passing through the launch polarizer box 40, the signal islaunched into the device under test, shown here as a fiber span 44.After the fiber span, the signal, now modified by the chromaticdispersion and PMD of the fiber span, enters a receiver 46 (part ofHewlett-Packard 83420A), which generates an electrical signal with avoltage proportional to the power in the optical signal. A networkanalyzer 48 (Hewlett-Packard 8753D) measures the difference between thephase of the signal provided by the oscillator 28 and the phase of theamplitude modulation of the received signal.

[0030] An example of measurements made using the apparatus shown in FIG.2 follows. Note that the detailed math for this general case will bediscussed later.

[0031] Four polarization states (linear polarization states at 0°, 60.4°and 120.6°, and a circular polarization state) were launched into a62-km optical fiber span having a known average PMD of 35 ps and achromatic dispersion of approximately +124 ps/nm at 1542 nm.

[0032]FIG. 3 shows the signal delays (vertical axis) for each inputpolarization as a function of light wavelength (horizontal axis).(Because the intrinsic signal delays through the 62-km fiber span areabout 300 μs, “relative” signal delays are plotted in FIG. 3 where aconstant delay has been subtracted from the actual measured delays.) Theoverall positive slope of the delay curves arises from the chromaticdispersion. The PMD of the fiber span causes the polarization dependenceof the signal delay at each wavelength, and the variations in thoserelative signal delay curves originate from the change of direction andmagnitude of {right arrow over (Ω)}(t)) as well as from underlyingchromatic dispersion.

[0033]FIG. 4 shows the relative polarization-averaged delay, τ₀ for thisfiber, determined from the data of FIG. 3. Here, τ₀ is the common, orpolarization-averaged, delay through the fiber span at each wavelengthcompared to the delay at the center wavelength, 1542.0 nm. In order toavoid “aliasing” and accurately measure the relative delays, themodulation frequency f_(m) must obey the condition: f_(m)<(2DΔλ)⁻¹. HereD is the dispersion of the optical element (ps/nm), and Δλ is the stepsize between adjacent wavelengths (nm). For FIG. 3, the step size Δλ was1 nm and the fiber's dispersion D was 124 ps/nm, so the modulationfrequency of 1 GHz obeyed the afore-mentioned condition.

[0034]FIG. 4 shows all the data necessary for the determination of thechromatic dispersion of the test fiber 44 (FIG. 2). Chromatic dispersionis the change in the intrinsic, or common (polarization-averaged) groupdelay τ₀ through an optical device with wavelength. For instance, ifmeasurements are taken at two wavelengths, λ₁ and λ₂, the chromaticdispersion D at the average of these wavelengths, (λ₁+λ₂ )/2, can beobtained from: $\begin{matrix}{{D\left\lbrack {\left( {\lambda_{1} + \lambda_{2}} \right)/2} \right\rbrack} = \frac{{\tau_{0}\left( \lambda_{2} \right)} - {\tau_{0}\left( \lambda_{1} \right)}}{\lambda_{2} - \lambda_{1}}} & (5)\end{matrix}$

[0035] As previously described, one method, according to the presentinvention, for determining τ₀ at a single wavelength comprises launchingtwo polarizations that are orthogonal in Stokes space (Ŝ_(A) and Ŝ_(−A))and measuring the corresponding mean signal delay for each, τ_(A) andτ_(−A).The common group delay is then $\begin{matrix}{\tau_{0} = {\frac{1}{2}\left( {\tau_{A} + \tau_{- A}} \right)}} & (6)\end{matrix}$

[0036] A simple example (equation 3) of two orthogonal polarizations arelinear polarizations with horizontal and vertical orientations, asindicated by the Stokes vectors, Ŝ₁ and Ŝ⁽⁻¹⁾, respectively. But, anypair of orthogonal polarizations will be sufficient.

[0037] In a more generalized method, launched are any four (known)non-degenerate input polarization launches {right arrow over (S)}_(i)(i=1, a, b, c) that span Stokes space and measurements are made of thecorresponding mean signal delays, τ_(g1), τ_(ga) τ_(gb), and τ_(gc).Using eq. 1, these mean signal delays can be expressed as$\begin{matrix}{\tau_{g1} = {\tau_{0} - {\frac{1}{2}{{\overset{\rightarrow}{s}}_{1} \cdot \overset{\rightarrow}{\Omega}}}}} & (7) \\{\tau_{ga} = {\tau_{0} - {\frac{1}{2}{{\overset{\rightarrow}{s}}_{a} \cdot \overset{\rightarrow}{\Omega}}}}} & (8) \\{\tau_{gb} = {\tau_{0} - {\frac{1}{2}{{\overset{\rightarrow}{s}}_{b} \cdot \overset{\rightarrow}{\Omega}}}}} & (9) \\{\tau_{gc} = {\tau_{0} - {\frac{1}{2}{{\overset{\rightarrow}{s}}_{c} \cdot \overset{\rightarrow}{\Omega}}}}} & (10)\end{matrix}$

[0038] When the polarization launch {right arrow over (s)}₁ is chosencorrectly such that there are no collinear pairs amongst {right arrowover (s)}_(a), {right arrow over (s)}_(b),and {right arrow over(s)}_(c), then {right arrow over (s)}_(i) can be expressed as asuperposition,

{right arrow over (s)} ₁=α₁ {right arrow over (s)} _(a)+β₁ {right arrowover (s)} _(b)+γ₁ {right arrow over (s)} _(c)   (11)

[0039] where the coefficients α₁, β₁, and γ₁ can be determined from therelations:

α₁ ={right arrow over (s)} ₁·({right arrow over (s)} _(b) ×{right arrowover (s)} _(c))/({right arrow over (s)} _(a)·({right arrow over (s)}_(b) ×{right arrow over (s)} _(c))),  (11a)

β₁ ={right arrow over (s)} ₁·({right arrow over (s)} _(a) ×{right arrowover (s)} _(c))/({right arrow over (s)} _(b)·({right arrow over (s)}_(a) ×{right arrow over (s)} _(c))),  (11b)

and γ₁ ={right arrow over (s)} ₁·({right arrow over (s)} _(a) ×{rightarrow over (s)} _(b))/({right arrow over (s)} _(c)·({right arrow over(s)} _(a) ×{right arrow over (s)} _(b))),  (11c)

[0040] or from the procedure involving eq. (16) to be described later.The above relations are obtained by forming the vector dot product ofeq. (11) with the appropriately chosen cross products.

[0041] A sum of eqn. (7) minus eqn. (8) times α₁ minus eqn. (9) times β₁minus eqn. (10) times γ₁ is taken to obtain:

τ_(g1)−α₁τ_(ga)−β₁τ_(gc)−γ₁τ_(gc)=τ₀(1−α₁−β₁−γ₁)  (12)

[0042] The common group delay τ₀ is then found from: $\begin{matrix}{\tau_{0} = {\frac{\tau_{g1} - {\alpha_{1}\tau_{ga}} - {\beta_{1}\tau_{gb}} - {\gamma_{1}\tau_{gc}}}{\left( {1 - \alpha_{1} - \beta_{1} - \gamma_{1}} \right)}\quad.}} & (13)\end{matrix}$

[0043] Note that one way of describing that the four launchpolarizations “span” Stokes space is to ensure that a tetrahedron isformed by the vertices defined by plotting the four input polarizationsin (normalized) Stokes space. (In general, the larger the volume of thistetrahedron, the more accurate the measurement.) Spanning Stokes spacealso means that not more than two of the polarizations can be collinear(in Stokes space).

[0044] In FIG. 4, the chromatic dispersion of the fiber corresponds tothe slope of the curve plotting τ₀ versus wavelength. For example, at awavelength of 1542.0 nm, the chromatic dispersion is +124 ps/nm.

[0045] As previously noted, because equation (1) contains four unknowns,four measurements are necessary. This is why, in the “more generalizedmethod” just described, four polarization launches are used. Conversely,if orthogonal polarizations are used, as previously explained, only twolaunches are needed. This follows because, with orthogonal launches, thepresence of PMD affects the delay of each signal by opposite, but equalamounts. That is, the deviations of the delays from τ₀ for the twolaunched signals have equal magnitude, but one signal is advanced whilethe other signal is retarded.

[0046] This is seen from equation 1. When changing a launch polarizationto an orthogonal polarization, the vector {right arrow over (s)} inequation 1 changes sign. That is, if one launched signal has vector{right arrow over (s)}₁ and a second launched signal has vector {rightarrow over (s)}₂, and if the two launches are orthogonal, then {rightarrow over (s)}₁=−{right arrow over (s)}₂. Thus, from equation 1, thedeviations of the delays, τ_(g), from τ₀ have equal magnitudes, butopposite signs. The PMD effects are thus cancelled in the measurements.

[0047] Instead of determining “moments” of the output signals frompulse-delay measurements, the signal delays, τ_(gi), can also beobtained (in approximation, i.e., to an acceptable degree of accuracy,e.g., 6% as described hereinafter) by observing phase shifts of themodulation of a sinusoidal amplitude modulated signal and benefitingfrom the precision of sensitive phase-detection techniques. (Note thatWilliams [Electron. Lett. Vol. 35, pp. 1578-1579, 1999] has usedsinusoidal modulation for the determination of the scalar DGD [Δτ].) Formost purposes, it suffices to only measure changes in τ_(gi) withpolarization and optical frequency and to not resolve the ambiguitypresented by the use of a signal with periodic modulation. For asinusoidal intensity-modulated signal with the assumption offrequency-independent PSP's and DGD, Eq. (1) becomes

tan ω_(m)(τ_(g)−τ₀)=−{right arrow over (q)}·{right arrow over(s)}tan(ω_(m)Δτ/2),  (14)

[0048] where ω_(m) is the angular modulation frequency. When the fourlaunch polarizations coincide with the Poincaré sphere axes, {rightarrow over (S)}₁, −{right arrow over (S)}₁, {right arrow over (S)}₂, and{right arrow over (S)}₃, Eq. (3) is still valid for sinusoidalmodulation:$\tau_{0} = {\frac{1}{2}{\left( {\tau_{g1} + \tau_{g{({- 1})}}} \right)\quad.}}$

[0049] In the more general case we can use any four non-degenerate inputpolarization launches {right arrow over (s)}_(i)(i=1, a, b, c ) thatspan Stokes space, where , {right arrow over (s)}₁=Ŝ₁ {right arrow over(s)}_(a)=(a₁, a₂, a₃ ), {right arrow over (s)}_(b)=(b₁, b₂, b₃), and{right arrow over (s)}_(c)=(c₁, c2, c3 ), and measure the correspondingdelays τ_(g1), τ_(ga), τ_(gb), and τ_(gc). We first express Ŝ₁, Ŝ₂, andŜ₃ in terms of {right arrow over (s)}_(a), {right arrow over (s)}_(b),and {right arrow over (s)}_(c),

Ŝ _(i)=α_(i) {right arrow over (s)} _(a)+β_(i) {right arrow over (s)}_(b)+γ_(i) {right arrow over (s)} _(c)(i=1, 2, 3),  (15)

[0050] where the coefficients α_(i), β_(i), and γ_(i) are obtained from$\begin{matrix}{{\begin{matrix}\alpha_{1} & \beta_{1} & \gamma_{1} \\\alpha_{2} & \beta_{2} & \gamma_{2} \\\alpha_{3} & \beta_{3} & \gamma_{3}\end{matrix}} = {\begin{matrix}a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{matrix}}^{- 1}} & (16)\end{matrix}$

[0051] Substituting Ŝ_(i)=α₁{right arrow over (s)}_(a)+β₁{right arrowover (s)}_(b)+γ₁{right arrow over (s)}_(c) into Eq. (14) gives

tanω_(m)(τ_(g1)−τ₀)=−{right arrow over (q)}·(α₁ {right arrow over (s)}_(a)+β₁ {right arrow over (s)} _(b)+γ₁ {right arrow over (s)}_(c))tan(ω_(m)Δτ/2)  (17)

[0052] leading to a transcendental equation for τ₀:

tanω_(m)(τ_(g1)−τ₀)=α₁tanω_(m)(τ_(ga)−τ₀)+β₁ tanω_(m)(τ_(gb)−τ₀)+γ₁tanω_(m)(τ_(gc)−τ₀). (18)

[0053] When solving for τ₀, a first trial value for τ₀ can be obtainedby linearizing Eq. (18), $\begin{matrix}{{\tau_{0} = \frac{\tau_{g1} - {\alpha_{1}\tau_{ga}} - {\beta_{1}\tau_{gb}} - {\gamma_{1}\tau_{gc}}}{1 - \alpha_{1} - \beta_{1} - \gamma_{1}}},} & (19)\end{matrix}$

[0054] which is equivalent to the earlier eq. (13) for the exactsolution of the pulse delay measurement.

[0055] Although the above procedure will yield any computationalaccuracy desired, it is often not necessary. For small modulationfrequencies, (ω_(m), we can approximate tan(x)≅x in Eq. (14), reducingto the earlier expressions (i.e. Eqs. (3) and (13)). These linearexpressions are valid for sinusoidal modulation to within 6% as long asω_(m)Δτ<π/4. For instance, for the peak DGD we observed here, 70 ps, 6%accuracy will be obtained for modulation frequencies, f_(m)=ω_(m)/2π,less than 1.8 GHz by using the linear expressions.

[0056] As above-explained, the inventive method involves multiplelaunchings of different polarized light states and determining the meansignal delays of the different polarization states. Basic techniques formeasuring the group delay of an optical device are known, see, e.g., Y.Horiuchi, et al., “Chromatic Dispersion Measurements of 4564 km OpticalAmplifier Repeater System,” Electronics Letters, Vol. 29, pp. 4-5, Jan.7, 1993. By comparing data taken at adjacent wavelengths, chromaticdispersion of optical devices can be accurately measured; suchmeasurements being accurate even with the presence of PMD.

What is claimed is:
 1. A method for measuring chromatic dispersion at anoptical frequency of an optical device comprising: launching, at aninput of said device, at each of at least two optical frequencies, twotest light signals having orthogonal polarization states, measuring, atan output of said device, the mean signal delay of each of said testsignals at each of said optical frequencies, and calculating thechromatic dispersion of said device as the change of the intrinsic groupdelay τ₀ with wavelength.
 2. A method according to claim 1 wherein themeasuring of the mean signal delays is accomplished in approximation bymeasuring phase delays of sinusoidally modulated signals.
 3. A methodaccording to claim 1 wherein the calculation is based upon the formula$\tau_{0} = {\frac{1}{2}\left( {\tau_{g1} + \tau_{g{({- 1})}}} \right)}$

where, for the test signals at each said optical frequency, τ₀ is theintrinsic group delay and τ_(g1) and τ_(g(−1)) are the mean signaldelays of the respective orthogonal light signals.
 4. A method accordingto claim 3 wherein the measurements are made at two wavelengths, λ₁ andλ₂, and the chromatic dispersion, D, at the average of said twowavelengths, is obtained based upon the formula:${D\left\lbrack {\left( {\lambda_{1} + \lambda_{2}} \right)/2} \right\rbrack} = \frac{{\tau_{0}\left( \lambda_{2} \right)} - {\tau_{0}\left( \lambda_{1} \right)}}{\lambda_{2} - \lambda_{1}}$


5. A method for measuring chromatic dispersion at a first opticalfrequency of an optical device comprising: launching, at an input ofsaid device, at each of at least two optical frequencies, fournon-degenerate light signals {right arrow over (s)}_(i)(i=1,a,b,c) thatspan Stokes space, measuring, at an output of said device, the meansignal delay τ_(gi)(i=1,a,b,c) of each of said test signals at each ofsaid optical frequencies, and calculating the chromatic dispersion ofsaid device as the change of the intrinsic group delay τ₀ withwavelength, where the calculation is based upon the formula:$\begin{matrix}{\tau_{0} = \frac{\tau_{g1} - {\alpha_{1}\tau_{g\quad a}} - {\beta_{1}\tau_{g\quad b}} - {\gamma_{1}\tau_{g\quad c}}}{1 - \alpha_{1} - \beta_{1} - \gamma_{1}}} & (13)\end{matrix}$

where the coefficients α₁, β₁, and γ₁ are determined from the relations:α₁ ={right arrow over (s)} ₁·({right arrow over (s)} _(b) ×{right arrowover (s)} _(c))/({right arrow over (s)} _(a)·({right arrow over (s)}_(b) ×{right arrow over (s)} _(c))),  (11a) β₁ ={right arrow over (s)}₁·({right arrow over (s)} _(a) ×{right arrow over (s)} _(c))/({rightarrow over (s)} _(b)·({right arrow over (s)} _(a) ×{right arrow over(s)} _(c))),  (11b) γ₁ ={right arrow over (s)} ₁·({right arrow over (s)}_(a) ×{right arrow over (s)} _(b))/({right arrow over (s)} _(c)·({rightarrow over (s)} _(a) ×{right arrow over (s)} _(b))),  (11c)
 6. A methodaccording to claim 5 wherein the measurements are made at twowavelengths, λ₁ and λ₂, and the chromatic dispersion, D, at the averageof said two wavelengths, is obtained based upon the formula:${D\left\lbrack {\left( {\lambda_{1} + \lambda_{2}} \right)/2} \right\rbrack} = \frac{{\tau_{0}\left( \lambda_{2} \right)} - {\tau_{0}\left( \lambda_{1} \right)}}{\lambda_{2} - \lambda_{1}}$


7. A method for measuring chromatic dispersion at a first opticalfrequency of an optical device comprising: launching, at an input ofsaid device, at each of at least two optical frequencies, fournon-degenerate, sinusoidal amplitude modulated, light signals {rightarrow over (s)}_(i)(i=1,a,b,c) that span Stokes space, where {rightarrow over (s)}₁=Ŝ₁, {right arrow over (s)}_(a)=(a₁,a₂,a₃), {right arrowover (s)}_(b)=(b₁,b₂,b₃ ), and {right arrow over (s)}_(c)=(c₁,c₂,c₃),measuring, at an output of said device, the mean signal delayτ_(gi)(i=1,a,b,c) of each of said test signals at each of said opticalfrequencies, and calculating the chromatic dispersion of said device asthe change of the intrinsic group delay τ₀ with wavelength, where thecalculation is based upon the formula: tanω_(m)(τ_(g1)−τ₀)=α₁tanω_(m)(τ_(ga)−τ₀)+β₁ tanω_(m)(τ_(gb)−τ₀)+γ₁ tanω_(m)(τ_(gc)−τ₀) wherethe coefficients α₁,β₁, and γ₁ are determined from: $\begin{matrix}{{\begin{matrix}\alpha_{1} & \beta_{1} & \gamma_{1} \\\alpha_{2} & \beta_{2} & \gamma_{2} \\\alpha_{3} & \beta_{3} & \gamma_{3}\end{matrix}} = {\begin{matrix}a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3}\end{matrix}}^{- 1}} & (16)\end{matrix}$


8. A method according to claim 7 wherein the measurements are made attwo wavelengths, λ₁ and λ₂, and the chromatic dispersion, D, at theaverage of said two wavelengths, is obtained based upon the formula:${D\left\lbrack {\left( {\lambda_{1} + \lambda_{2}} \right)/2} \right\rbrack} = \frac{{\tau_{0}\left( \lambda_{2} \right)} - {\tau_{0}\left( \lambda_{1} \right)}}{\lambda_{2} - \lambda_{1}}$